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Direct observation of magnon fractionalization in the quantum spin ladder

Thielemann, B.; Rüegg, C.; Rønnow, H.M.; Läuchli, A.M.; Caux, J.S.; Normand, B.; Biner, D.;Krämer, K.W.; Güdel, H.U.; Stahn, J.; Habicht, K.; Kiefer, K.; Boehm, M.; McMorrow, D.F.;Mesot, J.DOI10.1103/PhysRevLett.102.107204Publication date2009Document VersionFinal published versionPublished inPhysical Review Letters

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Citation for published version (APA):Thielemann, B., Rüegg, C., Rønnow, H. M., Läuchli, A. M., Caux, J. S., Normand, B., Biner,D., Krämer, K. W., Güdel, H. U., Stahn, J., Habicht, K., Kiefer, K., Boehm, M., McMorrow, D.F., & Mesot, J. (2009). Direct observation of magnon fractionalization in the quantum spinladder. Physical Review Letters, 102(10), 107204.https://doi.org/10.1103/PhysRevLett.102.107204

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https://doi.org/10.1103/PhysRevLett.102.107204https://dare.uva.nl/personal/pure/en/publications/direct-observation-of-magnon-fractionalization-in-the-quantum-spin-ladder(b5a73902-8cae-4c4a-9811-9cc17ea85f25).htmlhttps://doi.org/10.1103/PhysRevLett.102.107204

Direct Observation of Magnon Fractionalization in the Quantum Spin Ladder

B. Thielemann,1 Ch. Rüegg,2 H.M. Rønnow,3 A.M. Läuchli,4 J.-S. Caux,5 B. Normand,6 D. Biner,7 K.W. Krämer,7

H.-U. Güdel,7 J. Stahn,1 K. Habicht,8 K. Kiefer,8 M. Boehm,9 D. F. McMorrow,2 and J. Mesot1,3

1Laboratory for Neutron Scattering, ETH Zurich and Paul Scherrer Institute, CH-5232 Villigen, Switzerland2London Centre for Nanotechnology and Department of Physics and Astronomy, University College London,

London WC1E 6BT, United Kingdom3Laboratory for Quantum Magnetism, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

4Max Planck Institut für Physik komplexer Systeme, Nöthnitzerstrasse 38, D-01187 Dresden, Germany5Institute for Theoretical Physics, University of Amsterdam, 1018 XE Amsterdam, The Netherlands

6Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich, Switzerland7Department of Chemistry and Biochemistry, University of Bern, CH-3000 Bern 9, Switzerland

8BENSC, Helmholtz Centre Berlin for Materials and Energy, D-14109 Berlin, Germany9Institut Laue Langevin, 6 rue Jules Horowitz BP156, 38024 Grenoble CEDEX 9, France

(Received 18 December 2008; published 13 March 2009)

We measure by inelastic neutron scattering the spin excitation spectra as a function of applied magnetic

field in the quantum spin-ladder material ðC5H12NÞ2CuBr4. Discrete magnon modes at low fields in thequantum disordered phase and at high fields in the saturated phase contrast sharply with a spinon

continuum at intermediate fields characteristic of the Luttinger-liquid phase. By tuning the magnetic field,

we drive the fractionalization of magnons into spinons and, in this deconfined regime, observe both

commensurate and incommensurate continua.

DOI: 10.1103/PhysRevLett.102.107204 PACS numbers: 75.10.Jm, 64.70.Tg, 75.40.Gb, 78.70.Nx

Gapped quantum antiferromagnets (AFs) offer some ofthe most exotic states of electronic matter ever to beobserved, and as such have been the enduring focus ofintense theoretical and experimental interest. In an appliedmagnetic field, the threefold degeneracy of the triplet ex-citations is lifted and the triplet gap � is closed at a criticalfield Bc ¼ �=g�B. The properties of the states beyond thisquantum critical point (QCP) depend strongly on systemdimensionality [1], and in one dimension (1D), wherelong-range order is forbidden, the field-induced phase isexpected to be a spin Luttinger liquid (LL). In this state,spin-flip excitations fractionalize into spinons, elementaryS ¼ 1=2 entities, whose excitation spectrum is dramati-cally different from that of both ordered and truly quantumdisordered (QD) magnets.

Here we present the results of inelastic neutron scatter-ing (INS) measurements of the magnetic excitation spec-trum of a quantum spin ladder. By tuning the magneticfield, we access the full ladder phase diagram, whichincludes QD, LL, and fully saturated (FM) phases. INS isthe only experimental technique which can measure thefull momentum and energy dependence of the dynamicalsusceptibility, information which is essential to identifyunambiguously the nature of the quasiparticles in thesedifferent phases. We will show that both the QD phase(ladder magnetization m ¼ 0) and the FM phase (m ¼ 1)have well-defined magnon modes. In sharp contrast, thespectrum in the gapless LL state (0

[Qk and Ql, Figs. 1(a) and 1(c)] is shown to demonstratethe excellent one-dimensionality of the system. In fact, wehave found a systematic variation of order 30 �eV in theeffective ladder bandwidth [14], and subtract this interlad-der contribution to obtain the intrinsic parameters Jr andJl. Figures 1(d) and 1(e) show the INS intensity at Q ¼½1:5 0 0� (the band minimum), demonstrating a Zeemansplitting into three triplet components at finite field;Gaussian fits yield the energies shown in Fig. 1(f).

The spectrum changes dramatically above Bc: we find acontinuum of excitations extending over much of theBrillouin zone and up to energies of 0.8 meV. We considerfirst a field corresponding to half magnetic saturation (m ¼0:5, Fig. 2), where the ladder is equivalent to a gapless spinchain in zero field (below). Here the continuous spectrum

of spinon excitations [15] is bounded by �lðQhÞ ¼ @!ðQhÞand �uðQhÞ ¼ 2@!ðQh2 Þ, where @!ðQhÞ ¼ �Jj sinð2�QhÞj[16] with � a quantum renormalization factor which isdetermined exactly from the system geometry and inter-action parameters. In Fig. 2(a), it is clear that the com-mensurate m ¼ 0:5 spectrum is well described by such ashape. Its continuum nature is illustrated strikingly inhigh-statistics measurements taken at B ¼ 10:1 T: bothconstant-E [Figs. 2(b) and 2(d)] and constant-Q[Fig. 2(c)] scans show broad regions of continuous inten-sity, the latter extending from 0.15 meV (lower measure-ment limit) to 0.8 meV. We stress two important points.First, these data are taken well inside the LL regime, at atemperature significantly below the LL crossover, TLL[11], but above the boundary to 3D order induced by

0

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h 0 0]

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FIG. 1 (color online). Spin dynamicsin the QD phase (B< Bc) at T ¼50 mK. (a)–(c) Triplet dispersion at B ¼0 T from constant-Q scans along theladder axis and in the two perpendiculardirections. (d)–(e) Triplet excitations atQ ¼ ½1:5 0 0� in fields of 0 and 4 T.(f) Zeeman splitting of the triplet modesat Q ¼ ½1:5 0 0�. Solid lines are fits ex-plained in the text.

FIG. 2 (color online). Excitation spec-trum in the LL phase (TN < T ¼250 mK< TLL) at B ¼ 10:1 T (m �0:5) after subtraction of the zero-fieldbackground. (a) Measured (left) andsimulated (right) INS intensities. Solidlines mark the edges of the two-spinoncontinuum. (b),(d) Constant-E scanstaken along maxima of the transversestructure factor (insets: scan trajectoriesin white). (c) Constant-Q scan at Q ¼½0:5 0 0:61�. Black dashed lines in (b)–(d) are based on a �-function spinonspectrum [solid lines in panel (a)], redsolid lines and shading on a full contin-uum calculation.

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interladder coupling (TN;max ¼ 110 mK, [12,13]). Second,the spinon continuum studied in Figs. 2 and 3 arises onlyfrom fractionalization of the lowest triplet branch of theQD phase.

We observe a continuous evolution of the excitationspectrum as the magnetic field is tuned away from m ¼0:5. Figures 3(c)–3(f) show constant-E scans at B ¼ 7:9and 12:3 T (corresponding approximately to m ¼ 0:25and 0:75) which are essentially identical within the ex-perimental error. At E ¼ 0:4 meV, the decrease of mag-netic intensity compared to Fig. 2(b) between Qh ¼ 0:2and 0.8 is accompanied by additional weight around thezone boundaries (ZBs, which we define as Qh ¼ 0 and 1).For these values of m, field-induced shifts in spectralweight are less pronounced at E ¼ 0:2 meV. Because itis not possible by INS to follow the location of the zero-energy incommensurate point, instead we have mea-sured the magnetic signal at the ZB as a function of field[Figs. 3(g) and 3(h)], finding an increase in energy andintensity from m ¼ 0:5 to m ¼ 0 and 1.

When the magnetic field is increased beyond a secondQCP at Bs, the spins are fully aligned. The spectrumbecomes discrete again, as shown in Figs. 4(d) and 4(e),with elementary magnon excitations. We observe a 1Ddispersive band [Figs. 4(b) and 4(c)], whose width isvery similar to that measured in the QD phase. The ZBexcitation energy increases linearly with applied field[Fig. 4(a)], which allows the identification of Bs.

We use the magnon dispersion relations to deducethe exchange parameters of the system. BecauseðC5H12NÞ2CuBr4 is rather ‘‘strongly coupled’’ (Jr=Jl �4), high-order perturbative expansions are very effective,and here we follow the (3D) treatment of Ref. [17]. Themeasured triplet dispersion is dominated by the ladderterms: we obtain Jr ¼ 12:8ð1Þ K and Jl ¼ 3:2ð1Þ K [blackline in Fig. 1(b)]. From the fits to the linear Zeemansplitting [solid lines in Fig. 1(f)], the g factor for thisorientation is g ¼ 2:17ð3Þ, while Bc ¼ 6:8ð1Þ T. The twoapproaches agree perfectly, and are consistent with valuesof g and Bc determined by other techniques.Theoretically, the excitations of a field-polarized ladder

are gapped spin waves with dispersion relation �ðQhÞ ¼g�BðB� BsÞ þ Jl½1þ cosð2�QhÞ� [18]. A mean-fieldtreatment is exact here because all quantum fluctuationsare quenched. The red lines in Figs. 4(b) and 4(c) are fittedusing this expression: when the small 3D coupling term isremoved, we obtain Jr ¼ 13:1ð1Þ K, Jl ¼ 3:3ð1Þ K, andBs ¼ 13:6ð1Þ T. The fit to the ZB energy in Fig. 4(a) yieldsBs ¼ 13:6ð2Þ T.The LL continuum arises from the fractionalization of

spin-flip excitations (�Sz ¼ 1) into two elementary anddeconfined S ¼ 1=2 objects. Indeed, in a unified descrip-tion of the full phase diagram, the magnon excitations inthe QD and FM phases are bound states of these spinons,and the QCP at Bc may be regarded as a field-driven spinonbinding-unbinding or confinement-deconfinement transi-

FIG. 3 (color online). Incommensurate excitations in the LL phase. (a),(b) Spinon continua calculated for an S ¼ 1=2 XXZ chainwith anisotropy � ¼ 0:5 (see text) at m ¼ 0:25, 0.75 (a) and m ¼ 0:5 [(b), data shown also in Fig. 2(a) convolved with instrumentalresolution]; dashed white lines represent INS scans at E ¼ 0:2 and 0.4 meV. (c)–(f) Constant-E scans (trajectories as in Fig. 2(b) inset)for B ¼ 7:9 and 12:3 T. (g) Constant-Q scans at the ZB for two chosen magnetizations m between 0 and 0.5. (h) Summary of data forZB excitation energy as a function of m. Red lines are predictions based on the XXZ chain: solid from panel (a) and dashed (shown forcomparison) from panel (b).

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tion. The theoretical description for a ladder in the strong-coupling limit may be obtained by a mapping to the S ¼1=2 XXZ chain with anisotropy � ¼ 1=2 [19]. The low-energy sector of the ladder [i.e., the lowest Zeeman-splitbranch in Fig. 1(f)] is governed by

HXXZ¼X

i

Jlð~Sxi ~Sxiþ1þ ~Syi ~Syiþ1þ�~Szi ~Sziþ1Þ�beff ~Szi ; (1)

where the effective field beff ¼ 2bs B�ðBcþBsÞ=2Bs�Bc is such that�bs � beff � bs, with bs ¼ 32 Jl the saturation field for theXXZ chain. This effective model has spinon excitations inwhose dispersion J ¼ Jl and � ¼ 1:299 [16], whence thebounds �lðQhÞ and �uðQhÞ shown as solid lines in Fig. 2(a).Because ~Si in Eq. (1) is a composite of the two physicalspins on each ladder rung, the total INS cross sectioncontains a rung structure factor modulating the contribu-tions from longitudinal and transverse spin correlations[14]. All data in the LL phase were measured on themaxima of the transverse structure factor (insets inFig. 2), where the longitudinal contributions are zero.

We have calculated the transverse spin correlationfunction for all values of m following Ref. [20], and inFigs. 3(a) and 3(b) present the results for m ¼ 0:25, 0.75,and 0.5. In the XXZ-chain model, the spectral intensitiesare symmetric in m about m ¼ 0:5. Data taken at B ¼10:1 T correspond to m ¼ 0:48ð2Þ, while fields B ¼ 7:9and 12:3 T correspond, respectively, to m ¼ 0:24ð2Þ and0:72ð2Þ. The theoretical intensities were convolved withthe 4D instrumental resolution to obtain the global fitshown as red lines in Figs. 2(b)–2(d) and 3(c)–3(f), whilethe red lines in Figs. 3(g) and 3(h) are obtained directly.The shaded bands indicate the error bar in the experimentaldetermination of a single constant of proportionality validfor all fields, energies, and wave vectors. Their widthcombines the statistics of all our scans with uncertaintiesin the exact magnetization values at the chosen fields and in

the convolution procedure. The agreement is quantitativelyexcellent. We note, in particular, that only the smallestasymmetries between m< 0:5 and m> 0:5 may be dis-cerned in the data [comparing Fig. 3(c) with 3(e) andFig. 3(d) with 3(f)], and that the energy dependence ofthe intensity is described exactly [Fig. 2(c)].Measurements in the QD and FM phases provide two

independent and complementary determinations of Jr andJl. While the values of Bc and Bs deduced from these aremarginally smaller than from thermodynamic studies, thelatter were performed mostly on undeuterated samples andin different orientations. By exploiting the Q specificity ofINS, we have accounted for a 3% effect from interladdercoupling [14]. We then find that the intrinsic ladder legparameters in the two regimes are identical within theirerrors, but the rung parameters are not: this apparentmagnetostriction effect is of order 1%–2%. That such aphenomenon may occur is not surprising in a structurally‘‘soft’’ material of this nature [10]. In fact this discrepancyis the sum of all additional contributions, including anyother magnetoelastic terms or complex spin interactions.Our magnon dispersion analysis therefore quantifies thestatement that ðC5H12NÞ2CuBr4 is an excellent spin-laddersystem.A key property of the LL spectrum is the presence of a

zero-energy mode at an incommensurate wave vector 0<Qmin < 1=2 which changes systematically with field [21].However, the spectral weight at Qmin vanishes as E ! 0[Fig. 3(a)], precluding a direct measurement of the incom-mensurability. Instead we have presented indirect confir-mation of the theoretical prediction in the form of the field-tuned finite-E spectra and the ZB energy [Figs. 2 and 3].The question of the evolution of spectral weight in the LLis of particular interest in the context of the commensurate3D ordered phase which emerges at sufficiently low tem-perature [12,13]. At intermediate energies, we return to thequestion of the symmetry of the measured spectra about

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FIG. 4 (color online). Spin dynamicsin the FM phase. (a) Magnetic-field de-pendence of the ZB excitation energy.(b),(c) Magnon dispersions along Ql andQh. (d),(e) Constant-E scans at B ¼14:8 T and T ¼ 50 mK with Gaussianfits. Solid lines in panels (a)–(c) are fitsto the dispersion, while dashed linesallow extraction of Bs.

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m ¼ 0:5: physical effects arising due to departures fromstrict strong coupling, and from the higher triplet branches,are expected to cause some asymmetry in intensities, butthese are clearly extremely small for energies 0:5Jl < E <2:5Jl. At high energies, it remains to address, both experi-mentally and theoretically, the nature of the higher spinoncontinua expected from the upper two triplet branches.

In summary, we have performed a comprehensive INSinvestigation of the magnetic excitation spectrum inðC5H12NÞ2CuBr4, a spin-ladder compound whose energyscales are perfectly suited to systematic studies in labora-tory fields. We observe the presence of a broad continuumof spinon excitations in the intermediate, Luttinger-liquid(LL) phase, which is starkly different from the discrete(magnon) excitations measured below the critical field(QD phase) and above saturation (FM phase). From theQD and FM results, we extract the ladder parameters withunprecedented accuracy, demonstrating directly that eventhe sum of all other effects beyond the ‘‘nearly ideal’’Hamiltonian of weakly coupled ladders falls below the2% level. The spinon continuum proves the occurrenceof field-induced fractionalization into elementary S ¼1=2 entities as the system enters the LL regime. An ex-cellent, fully quantitative description of the incommensu-rate continua measured at all fields is obtained from aneffective chain model for a ladder with the coupling ratioJr=Jl � 4 of ðC5H12NÞ2CuBr4.

This project was supported by the Swiss NationalScience Foundation (Division II and the NCCR MaNEP),the Royal Society, EPSRC, and FOM. This work is based

partially on experiments performed at the Swiss spallationneutron source, SINQ, at the Paul Scherrer Institute.

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